This paper explores the possibilities for such a mission. It’s not that far fetched technically. But realistically, I don’t think it would happen, even if the object looked more interesting than it currently does.

https://arxiv.org/abs/1711.03155

Wasn't there a practical design using atomic bombs to propel a spacecraft? Alien artifact would certainly justify the budget and risk. Though if a country determined it was alien, they'd probably not share it until their craft was well on the way.

If it’s traveling too fast for us to intercept it, it’s location is irrelevant.

If our craft can accelerate fast enough to intercet it, it’s position is still irrelevant since we know it’s trajectory.

You can slingshot off of anything in any direction if you can compute your exit vector correctly. A couple of loops between the Earth and Moon and you could pick up enough speed to get anywhere you want.

The trouble is computing these trajectories is pretty tricky and if you get even one manouver wrong you crater or burn up in the atmosphere.